Master equation for the degree distribution of a Duplication and Divergence network

TL;DR

This paper derives a master equation for the degree distribution in Duplication-Divergence networks, analyzing how mutation rates influence network topology and providing a mathematical framework validated by simulations.

Contribution

It introduces a novel master equation approach for Duplication-Divergence network growth, revealing how mutation rates affect degree distribution and network density.

Findings

Degree distribution converges if total mutation rate > 0.5

Asymptotic degree distribution depends on mutation rate differences

Simulation results agree with analytical predictions

Abstract

Network growth as described by the Duplication-Divergence model proposes a simple general idea for the evolution dynamics of natural networks. In particular it is an alternative to the well known Barab\'asi-Albert model when applied to protein-protein interaction networks. In this work we derive a master equation for the node degree distribution of networks growing via Duplication and Divergence and we obtain an expression for the total number of links and for the degree distribution as a function of the number of nodes. Using algebra tools we investigate the degree distribution asymptotic behavior. Analytic results show that the network nodes average degree converges if the total mutation rate is greater than 0.5 and diverges otherwise. Treating original and duplicated node mutation rates as independent parameters has no effect on this result. However, difference in these parameters results in a slower rate of convergence and in different degree distributions. The more different these parameters are, the denser the tail of the distribution. We compare the solutions obtained with simulated networks. These results are in good agreement with the expected values from the derived expressions. The method developed is a robust tool to investigate other models for network growing dynamics.